Some Results on Secant Varieties Leading to a Geometric Flip Construction

We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality statements about SecX and also to obtain information about linear systems on the blow up of projective space along a variety X. We use these results to geometrically construct, for varieties of arbitrary dimension, a flip first described in the case of curves by M. Thaddeus via Geometric Invariant Theory, and to give some examples of vanishing theorems for the cohomology of powers of ideal sheaves.